1. Field of the Invention
The present invention relates to a gyrator, and more particularly, to a gyrator with feedback resistors.
2. Description of the Prior Art
Gyrators are one of many kinds of electronic circuits for converting impedance. For example, a gyrator has the capability to make an inductance circuit behave like a capacitance circuit. In the design of a continuous time filter, an integrator or a gyrator is frequently used for converting impedance. In fact, two integrators can be electrically connected in a loop to form a gyrator.
Please refer to FIG. 1 and FIG. 2. FIG. 1 is a circuit diagram of a Nauta cell gyrator NCG having a gyrator core GCi disclosed in U.S. Pat. No. 6,490,706. FIG. 2 is an equivalent circuit diagram of the gyrator NCG. In the design of a filter of the prior art, at least a filter section FSTi comprises the gyrator NCG. In FIG. 2, a relation between an input current I1 and an output voltage V2 is I1=−gm*V2, and another relation between an input voltage V1 and an output current is I2=gm*V1, where −gm is a negative trans-conductance of the gyrator NCG and gm is a positive trans-conductance.
In addition to the gyrator core GCi, the gyrator NCG shown in FIG. 1 further comprises two common mode feedback sections CMIi and CMOi. The gyrator core GCi comprises four inverters GI1i, GI2i, GI3i, and GI4i electrically connected between two input ends i_1 and i_2 and two output ends o_1 and o_2. The common mode feedback section CMIi comprises two non-reverse series connection inverter sets, each of which comprises an inverter CMI1 and a short-circuited inverter CMI2. The common mode feedback section CMOi also comprises two non-reverse series connection inverter sets, each of which comprises an inverter CMO1 and a short-circuited inverter CMO2.
Please refer to FIG. 3, which is an equivalent circuit diagram of an inverter (such as the inverter CMI1, or the inverter CMO1) of the gyrator NCG shown in FIG. 1. The inverter has an admittance matrix equal to
                                          Y            inv                    =                      [                                                                                                      y                      i                                        +                                          y                      f                                                                                                            -                                          y                      f                                                                                                                                                              y                      m                                        -                                          y                      f                                                                                                                                  y                      o                                        +                                          y                      f                                                                                            ]                          ,                            (                  eq          .                                          ⁢          1                )            
where yi=sci is an input admittance, yo=go+sco is an output admittance, yf=scf is a trans-admittance from an output end to an input end, and ym is a trans-admittance from the input end to the output end.
Based on eq.1, an admittance matrix of the gyrator core GCi can be derived as
      Y    core    =      [                                                      y              l                        +                          y              f                                                            -                          y              f                                                0                                                    y              m                        -                          y              f                                                                                      y              m                        -                          y              f                                                                          y              l                        +                          y              f                                                            -                          y              f                                                0                                      0                                                    y              m                        -                          y              f                                                                          y              l                        +                          y              f                                                            -                          y              f                                                                        -                          y              f                                                0                                                    y              m                        -                          y              f                                                                          y              l                        +                          y              f                                            ]  
(without losing the generality, the gyrator NCG is assumed to comprise identical inverters GI1i, GI2i, GI3i, GI4i, CMI1, CMI2, CMO1, and CMO2), and an admittance matrix of the common mode feedback section can be derived as
            Y      CM        =          [                                                                  2                ⁢                                  y                  l                                            +                              y                m                                                          0                                                              y                m                            -                              2                ⁢                                  y                  f                                                                          0                                                0                                                              2                ⁢                                  y                  l                                            +                              y                m                                                          0                                                              y                m                            -                              2                ⁢                                  y                  f                                                                                                                        y                m                            -                              2                ⁢                                  y                  f                                                                          0                                                              2                ⁢                                  y                  l                                            +                              y                m                                                          0                                                0                                                              y                m                            -                              2                ⁢                                  y                  f                                                                          0                                                              2                ⁢                                  y                  l                                            +                              y                m                                                        ]        ,
where yi is equal to yi+yf+yo. Accordingly, an admittance matrix of the gyrator NCG can be derived as
                              Y          gyr                =                              [                                                  ⁢                                                                                                      3                      ⁢                                              y                        l                                                              +                                          y                      m                                        +                                                                                        -                                          y                      f                                                                                                                                  y                      m                                        -                                          2                      ⁢                                              y                        f                                                                                                                                                        y                      m                                        -                                          y                      f                                                                                                                                        y                    f                                                                                                                                                                                                                                                                                                                                                                                                                  y                      m                                        -                                          y                      f                                                                                                                                  3                      ⁢                                              y                        l                                                              +                                          y                      m                                        +                                                                                        -                                          y                      f                                                                                                                                  y                      m                                        -                                          2                      ⁢                                              y                        f                                                                                                                                                                                                                            ⁢                    `                                                                                        y                    f                                                                                                                                                                                                                                                                                                                          y                      m                                        -                                          2                      ⁢                                              y                        f                                                                                                                                                        y                      m                                        -                                          y                      f                                                                                                                                  3                      ⁢                                              y                        l                                                              +                                          y                      m                                        +                                                                                        -                                          y                      f                                                                                                                                                                                                                                                                                                                        y                    f                                                                                                                                                                                                            -                                          y                      f                                                                                                                                  y                      m                                        -                                          2                      ⁢                                              y                        f                                                                                                                                                        y                      m                                        -                                          y                      f                                                                                                                                  3                      ⁢                                              y                        l                                                              +                                          y                      m                                        +                                                                                                                                                                                                                                                                                                                                                                                            y                    f                                                                        ⁢                                                  ]                    .                                    (                  eq          .                                          ⁢          2                )            
Under an assumption that the applied signal of the gyrator NCG is differential, that is Vi1=−Vi2, Vo1=−Vo2, Ii1=−Ii2, and Io1=−Io2, where Vi1, Vi2, Vo1 and Vo2 are four voltages on two input ends i_1 and i_2 and two output ends o_1 and o_2 respectively, and Ii1, Ii2, Io1, Io2 four currents flowing through the input ends i_1 and i_2 and the output ends o_1 and o_2 respectively, eq.2 can be simplified as
                              Y          gyr                =                              [                                                                                                      3                      ⁢                                                                                          ⁢                                              (                                                                              y                            i                                                    +                                                      2                            ⁢                                                          y                              f                                                                                +                                                      y                            o                                                                          )                                                              +                                          Δ                      ⁢                                                                                          ⁢                                              y                        m                                                                                                                                  -                                          y                      m                                                                                                                                        y                    m                                                                                                              3                      ⁢                                                                                          ⁢                                              (                                                                              y                            i                                                    +                                                      2                            ⁢                                                          y                              f                                                                                +                                                      y                            o                                                                          )                                                              +                                          Δ                      ⁢                                                                                          ⁢                                              y                        m                                                                                                                  ]                    .                                    (                  eq          .                                          ⁢          3                )            
If YI is defined to be equal to 3(yi+2yf+yo), eq.3 can be further simplified as
                                          Y            gyr                    =                      [                                                                                                      Y                      l                                        +                                          Δ                      ⁢                                                                                          ⁢                                              y                        m                                                                                                                                  -                                          y                      m                                                                                                                                        y                    m                                                                                                              Y                      l                                        +                                          Δ                      ⁢                                                                                          ⁢                                              y                        m                                                                                                                  ]                          ,                            (                  eq          .                                          ⁢          4                )            
where Δym is a difference between trans-admittances ym.
In U.S. Pat. No. 6,490,706, a channel delay effect is taken into consideration, that is ym=gme−Sτ, and τ=cm/gm, where τ is an effective channel delay of the gyrator core GCi, gm is an effective gyrating constant, and cm is an effective trans-capacitance. The gyrator NCG has to function in a stable condition: g*c is not smaller than gm*cm, where g is an effective conductive loading of the gyrator GCi, and c is an effective capacitive loading.
However, the above stable condition is applied to a specific case only.
Moreover, since an integrator of the gyrator NCG has a DC gain equal to A0=gm/g, and g is proportional to I/L, and gm is proportional to I/Vod, where I is a bias current of the inverter, L is a channel length, and Vod is an overdrive voltage, if gm is kept constant, A0 can be increased through a decrease in I, or another increase in L so as to decrease g. However, in the meantime of the decreasing 1, Vod is decreased accordingly. Therefore, the gyrator has a poor linearity. On the other hand, cm is increased as L is increased, and the stability of the gyrator NCG will be reduced further.